Griffith used two strains of Pneumococcus (which infects mice), a S (smooth) and a R (rough) strain. The S strain covers itself with a polysaccharide capsule that protects it from the host's immune system, resulting in the death of the host, while the R strain doesn't have that protective capsule and is defeated by the host's immune system.
In his experiment, bacteria from the S strain were killed by heat, and their remains were added to R strain bacteria. It turned out that the formerly harmless R strain now was able to kill its host. It had been transformed into the lethal S strain, obviously by a transforming principle that was somehow part of the dead S strain bacteria.
Today, we know that the DNA of the S strain bacteria had survived the heating process, and was taken up by the R strain bacteria. The S strain DNA contains the genes that form the protective polysaccharide capsule. Equipped with this gene, the former R strain bacteria were now protected from the host's immune system and could kill it.